The Naming of Musical Notes, Part I
How many different notes are there in an octave? What about note names? The answer to the second is very interesting and this is part one of an exploration of that question.
Bach's Well-Tempered Clavier (henceforce WTC) consists of two books each containing prelude and fugue pairs in 24 different keys. Why 24? Well, if you look at a keyboard, you'll see there are 12 notes in the octave. Allowing for both major and minor keys we therefore have 24 major+minor keys to choose from and Bach wrote a prelude and fugue in each key in each book of the WTC.
But if we look at the key signature, it tells a different story. A key signature may consist of 1-7 sharps or 1-7 flats or nothing at all. Allowing for both major and minor keys that gives us 30 different keys.
Here are the 15 major keys that the key signature gives us (with the ones Bach uses in WTC in bold):
C# F# B E A D G C F Bb Eb Ab Db Gb Cb
The corresponding relative minors (with the ones Bach uses in WTC in bold again) are:
A# D# G# C# F# B E A D G C F Bb Eb Ab
Notice that the extra keys possible but unused by Bach are enharmonic with keys that are used. Db major is enharmonic with C# major, Gb major with F# major and Cb major with B major. Similarly A# minor is enharmonic with Bb minor, D# minor with Eb mintor and Ab minor with G# minor. That is not to say that Db major is the same as C# major—for one they have different key signatures and the names of each degree of the scale is different (more on that later). They may even sound different depending on the tuning system used.
But this explains why Bach wrote in 24 major+minor keys, even though notation provided him with 30—he avoided enharmonic duplicates.
But this isn't the whole story. Notice that:
- not all enharmonic spellings are accounted for even in the 15 major and 15 minor keys expressible in the key signature
- the major list and minor list differ
The reason behind these two facts will be the subject of the next part.
Comments (4)
James Tauber on Sept. 15, 2005:
That said, I don't believe anything I've said in the post above is altered in any way by whether one is using partials from the harmonic series or from, say, equal temperament.
Multipass on June 3, 2006:
Nice site though.
T Reeves on Sept. 6, 2006:
Well-Tempered Clavier is a translation of the German Wohl-Temperierte Klavier, so Clavier is a reasonable spelling. BTW, German's and American's don't like unnecessary apostrophe's either!
Found your site via Ars Mathematica. Look forward to reading your study of the Poincare Conjecture and its proof.
PS The Quoniam from Bach's B minor Mass rocks!
Last Modified: Aug. 30, 2005
Author: jtauber
Jim Plamondon on Sept. 8, 2005:
The music theory (scales, chords, etc.) of cultures whose instruments produce inharmonic spectra often use different scales which provide a better fit with those timbres. For example, those cultures whose musical instruments involve (inharmonic) vibrating bars whose sounds are "processed" by (harmonic) resonating chambers -- such as Thai and African cultures whose primary instruments are marimba-like -- tend to use a scale similar to 7-tone equal-temperament, which fits the timbre of these instruments very well.
In both Western and non-Western musicial cultures, tuning/scales and instruments co-evolve to maximize the alignment between spectral peaks (partials) and scale steps. With harmonic timbres, that leads to Just Intonation; with marimba-like instruments, it leads to 7-tone equal temperament. (Check out Bill Sethares' excellent book "Tuning, Timbre, Spectrum, Scale" for arecent and complete discussion of this issue.)
Ina any case, then, the "number of musical notes per octave" and hence the naming of notes is dependent on the spectrum of the sound on which one's music theory is based.